Lesson Plan | Traditional Methodology | First Degree Inequality

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to establish a clear and detailed understanding of what linear inequalities are, how to solve them, and how to apply these concepts to practical problems. This creates a solid foundation for students to confidently advance in the study of inequalities and their applications.

Main Objectives

1. Describe what linear inequalities are and their basic properties.

2. Demonstrate how to solve linear inequalities step by step.

3. Apply the concepts learned to practical problems.

Introduction

Duration: (10 - 15 minutes)

🎯 Purpose: The purpose of this stage is to establish a solid foundation and connect the lesson's theme with practical situations in the students' daily lives. By doing this, we aim to capture students' interest and prepare them for a deeper understanding of linear inequalities.

Context

💡 Initial Context: Start the class by asking students if they have ever faced situations where they needed to choose between two or more options, like deciding between going to the movies or studying for a test. Explain that, just like in everyday life, in Mathematics we often need to make decisions based on specific conditions. Linear inequalities are mathematical tools that help us represent and solve these conditions clearly and objectively.

Curiosities

🌟 Curiosity: Did you know that inequalities are widely used in various fields of knowledge? For example, in economics, they are used to determine the feasibility of investments; in engineering, to ensure that structures are safe; and even in computer programming, to create efficient algorithms. Understanding inequalities can open doors to various careers and practical applications in daily life.

Development

Duration: (40 - 45 minutes)

🎯 Purpose: The purpose of this stage is to provide students with a comprehensive and detailed understanding of linear inequalities. By addressing the definition, properties, solving methods, and practical examples, the goal is to ensure that students acquire solid skills to solve inequalities and apply them in various contexts. Additionally, the proposed questions will serve to consolidate knowledge and allow guided practice.

Covered Topics

1. 📌 Definition of Linear Inequality: Explain that a linear inequality is a mathematical expression that involves a variable, usually represented by 'x', and uses inequality signs (>, <, ≥, ≤). The general form is ax + b > c, where 'a', 'b', and 'c' are real numbers and 'a' ≠ 0. 2. 📌 Basic Properties of Inequalities: Detail that inequalities follow some important properties, such as adding and subtracting equal members on both sides of the inequality, and multiplying or dividing both sides by a positive number. Highlight that when multiplying or dividing by a negative number, the direction of the inequality sign must be inverted. 3. 📌 Solving Linear Inequalities: Demonstrate, step by step, how to solve an inequality. Use the example 2x - 4 > 6. Add 4 to both sides, resulting in 2x > 10. Next, divide both sides by 2, obtaining x > 5. 4. 📌 Graphical Representation of Solutions: Explain how to represent the solution of an inequality on a number line. Show that for x > 5, the solution is represented by all numbers greater than 5, not including 5 itself (open circle). 5. 📌 Practical Problems: Provide examples of practical problems that can be solved using linear inequalities. Use everyday situations, such as determining the minimum number of products that need to be sold to achieve a specific profit.

Classroom Questions

1. 1. Solve the inequality 3x + 7 ≤ 16. What is the solution set? 2. 2. Determine the solution of the inequality -2x + 5 < 1 and represent it on the number line. 3. 3. A cinema sells tickets for R$ 15.00 each. If the fixed costs are R$ 200.00 and the variable costs are R$ 5.00 per ticket, how many tickets need to be sold for the cinema to make a profit?

Questions Discussion

Duration: (20 - 25 minutes)

🎯 Purpose: The purpose of this stage is to review and consolidate the content covered in the lesson, ensuring that students understand the solutions to the practical questions and can apply the concepts of linear inequalities in various situations. The detailed discussion of the answers and student engagement through questions and reflections aim to reinforce learning and promote a deeper understanding of the topic.

Discussion

• 📚 Discussion of the Questions:

• 1. Question 1: Solve the inequality 3x + 7 ≤ 16. What is the solution set?
• • Step 1: Subtract 7 from both sides: 3x + 7 - 7 ≤ 16 - 7, resulting in 3x ≤ 9.
• • Step 2: Divide both sides by 3: x ≤ 3.
• • Solution Set: {x | x ≤ 3}.
• 2. Question 2: Determine the solution of the inequality -2x + 5 < 1 and represent it on the number line.
• • Step 1: Subtract 5 from both sides: -2x + 5 - 5 < 1 - 5, resulting in -2x < -4.
• • Step 2: Divide both sides by -2 and invert the inequality sign: x > 2.
• • Representation on the Number Line: An open circle at x = 2 and all values to the right.
• 3. Question 3: A cinema sells tickets for R$ 15.00 each. If the fixed costs are R$ 200.00 and the variable costs are R$ 5.00 per ticket, how many tickets need to be sold for the cinema to make a profit?
• • Step 1: Define the inequality: 15n > 200 + 5n, where n is the number of tickets.
• • Step 2: Subtract 5n from both sides: 15n - 5n > 200, resulting in 10n > 200.
• • Step 3: Divide both sides by 10: n > 20.
• • Number of Tickets: The cinema needs to sell more than 20 tickets to make a profit.

Student Engagement

1. ❓ Student Engagement: 2. 1. How would you represent the solution of the inequality x ≥ -3 on the number line? 3. 2. What is the difference between solving an equation and an inequality? Give examples. 4. 3. Why does the inequality sign invert when we multiply or divide by a negative number? 5. 4. Imagine you are organizing an event and need to ensure that the number of participants exceeds 50 to cover costs. How would you write this condition as an inequality? 6. 5. In what other practical everyday situations do you think inequalities can be applied?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to review and consolidate the main points covered in the lesson, reinforcing students' understanding of the topic and highlighting the importance and practical applications of linear inequalities. This ensures that students leave the class with a clear and complete view of the content.

Summary

• Definition of a linear inequality as an expression involving a variable and using inequality signs.
• Basic properties of inequalities, including addition, subtraction, multiplication, and division, and the inversion of the inequality sign when multiplying or dividing by a negative number.
• Steps to solve an inequality, exemplifying with 2x - 4 > 6.
• Graphical representation of solutions on a number line.
• Application of concepts to practical problems, such as determining the minimum quantity of products to achieve a specific profit.

The lesson connected theory with practice by demonstrating how linear inequalities can be solved and applied in everyday situations, such as financial planning and decision-making based on specific conditions, making the content more tangible and relevant for students.

Understanding linear inequalities is crucial for various everyday situations, from managing personal finances to solving complex problems in areas like economics, engineering, and programming. Additionally, inequalities are essential tools for making informed and efficient decisions.